I maintain an product's Facebook Page in English, but many of the customers are Japanese, so I would like to provide a Japanese version of the page.
What is the best way to do so?
I hope there is an alternative to creating a separate page.
No, you can just localize you app.
Almost all dev frameworks themself provide support for localization besides the Facebook API.
When posting to FB you can click on Public button (before you hit share) and then Location/ Language and then choose the Japan as country when posting just for them.